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author | Manuel Eberl <eberlm@in.tum.de> |

Wed, 13 May 2020 13:00:03 +0200 | |

changeset 72061 | b9e4e15d4438 |

parent 72060 | 96fa0f8072f9 |

generalised pigeonhole principle in HOL-Library.FuncSet

--- a/src/HOL/Library/FuncSet.thy Wed May 13 12:55:33 2020 +0200 +++ b/src/HOL/Library/FuncSet.thy Wed May 13 13:00:03 2020 +0200 @@ -686,4 +686,35 @@ by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product) qed +subsection \<open>The pigeonhole principle\<close> + +text \<open> + An alternative formulation of this is that for a function mapping a finite set \<open>A\<close> of + cardinality \<open>m\<close> to a finite set \<open>B\<close> of cardinality \<open>n\<close>, there exists an element \<open>y \<in> B\<close> that + is hit at least $\lceil \frac{m}{n}\rceil$ times. However, since we do not have real numbers + or rounding yet, we state it in the following equivalent form: +\<close> +lemma pigeonhole_card: + assumes "f \<in> A \<rightarrow> B" "finite A" "finite B" "B \<noteq> {}" + shows "\<exists>y\<in>B. card (f -` {y} \<inter> A) * card B \<ge> card A" +proof - + from assms have "card B > 0" + by auto + define M where "M = Max ((\<lambda>y. card (f -` {y} \<inter> A)) ` B)" + have "A = (\<Union>y\<in>B. f -` {y} \<inter> A)" + using assms by auto + also have "card \<dots> = (\<Sum>i\<in>B. card (f -` {i} \<inter> A))" + using assms by (subst card_UN_disjoint) auto + also have "\<dots> \<le> (\<Sum>i\<in>B. M)" + unfolding M_def using assms by (intro sum_mono Max.coboundedI) auto + also have "\<dots> = card B * M" + by simp + finally have "M * card B \<ge> card A" + by (simp add: mult_ac) + moreover have "M \<in> (\<lambda>y. card (f -` {y} \<inter> A)) ` B" + unfolding M_def using assms \<open>B \<noteq> {}\<close> by (intro Max_in) auto + ultimately show ?thesis + by blast +qed + end